The problem of the commons

 

… the friends establish farming business…

 

It is spring. William and Harold are now released from jail and they both are going to graze goats in the summer on a common green. In the autumn, they are going to sell their goats. Their problem now is to decide how many goats they should graze. This problem is known since Hume (1739) and it is called the problem of the commons.

 

… what is the value of a goat for a farmer? …

 

The more a goat has grass the better it survives. If there are only a few goats on the green, adding one more does not harm the ones already grazing. But, if there are many goats, adding one more is harmful for all the goats and the value of a goat decreases remarkably.

 

… representation as a game …

 

Text Box: • Players: Harold and William • Strategy for each player: The number of the goats he owns. • The payoff for a player: His value of owning the goats; it is the total selling price minus the total cost of the goats.

 

… the solution of the game …

 

Denote the number of goats for Harold by gH and for William by gW. Assume that the goats are continuously divisible, i.e., gH and gW are real numbers instead of being integer values only.

 

Recall that the solution to the prisoners’ dilemma game was defined by using the players’ best response strategies. Similarly we can define the Nash equilibrium  here. Let us look at the following figure:

 

 

 

In the figure, there are some contours of the players’ payoff functions. Along a contour a player’s payoff is constant. It is assumed that the value of the inner contour is always greater than that of the outer contour. Let us first check if the point  in the figure is the Nash equilibrium solution of the game. If Harold decides to play , then William chooses his best response strategy and maximises his payoff on the black slashed vertical line. He chooses , where the vertical line touches one of his contours. This is because for any other choice, like , Harold’s payoff is smaller. Likewise,  maximises Harold’s payoff on the black slashed horizontal line corresponding to William’s choice . By definition the pair  is the Nash equilibrium of the game. In the following it is shown that the Nash equilibrium can be found by computing the intersection of the players’ best response curves.

 

    … the solution by using mathematics …

 

The payoff function for Harold is uH(gH,gW)=[P(gH+gW)-c]gH, where c is the cost of buying and caring for and P(gH+gW) is the selling price of a goat per goat. Adding one more goat to the green harms the rest more if there are many goats than if there were only a few goats on the green. This means a bigger drop for the selling price per goat in the former case. Therefore, the shape of the function P as a function of the total number of goats G, is as shown by the blue curve in the figure below. Here Gmax is the carrying capacity of the green.

 

 

William’s payoff is similar to that of Harold but with gH and gW interchanged. For simplicity, approximate the blue curve in the figure by the red line, i.e., assume that P(gH+gW)=(Gmax-gH-gW) if gH+gW≤Gmax and P=0 otherwise. Then uH(gH,gW)=(Gmax-gH-gW) gH and uW(gH,gW)=(Gmax-gH-gW) gW.

 

The Nash equilibrium can now be computed using the best response curves. For Harold it is defined as in the prisoners’ dilemma game so that it gives him the best response for any choice gW of William. This best response function is denoted by RH(gW), and it satisfies

              

 

The best response curves gH=RH(gW) and gW=RW(gH), that appear to be linear in this example, are shown by the red lines in the figure below.

 

 

Since  is Harold’s best response to William’s Nash equilibrium strategy , and vice versa, the Nash equilibrium satisfies

              

 

i.e., it is defined by the intersection of the two best response functions.

 

       … Pareto optimal solutions …

 

Pareto optimal solutions are defined in terms of Pareto optimal outcomes.

 

Text Box: By definition, we say that is a Pareto optimal solution if any other strategy pair gives a worse outcome for at least one of the players.

 

The definition implies that the Pareto optimal solutions are defined by the points of tangency of the players’ payoff contours. One such point is illustrated in the figure below. This result can be verified easily by considering any other point, which is worse for either one of the players or both of them.

 

 

In the Nash equilibrium, Harold and William overutilize the common green because they compete about it. Therefore, the Nash equilibrium is sometimes called competitive equilibrium. If you look at the figure you can see that there are plenty of Pareto optimal points indicated by the green line segment. The bolded part of the line segment presents those Pareto optimal points which give better payoffs for the players than the Nash equilibrium because the inner contours give better payoff for the players. The phenomenon was called prisoners’ dilemma in our previous example. Now it is called the tragedy of commons.

 

Further reading

 

·        Nash, J. (1951). “Non Cooperative Games”. Annals of Mathematics, 54, 286-95. Nash defines the Nash equilibrium.

·        Luce, R., and H. Raiffa (1957). Games and Decisions. John Wiley & Sons. A book on different types of games including co-operative game theory.

·        Gibbons, R. (1992).  A Primer in Game Theory. Prentice Hall. An easy introduction to game theory.

·        Hume, D. (1739). Treatise of Human Nature. Reprint. London: J.M. Dent (1952). An early book on social philosophy presenting the tragedy of the commons.

·        Hardin, G. (1968). “Tragedy of Commons”. Science, 162, 1243-1248. Hardin describes the problem of the commons in the society from the quantitative perspective.